Vecteurs analytiques réduits et analyticité. (Reduced analytic vectors and analyticity) (Q1174987)

Let \(P\) be a differential operator of order \(m\) with analytic coefficients in an open set \(\Omega\) of \(R^ n\). A reduced analytic vector, as refered to in the title, is related to a strictly increasing sequence \((k_ j)_{j\in \mathbb{N}}\) of integers \(k_ j\in \mathbb{N}\) and defined as a \(C^ \infty\)-function \(u\) in \(\Omega\) satisfying, for every compact set \(K\) of \(\Omega,\) \(\sup_{j\in \mathbb{N}}\| P^{k_ j}u\|_{L_ 2(K)}C_ K^{k_ j}/(mk_ j)!<\infty\) for some constant \(C_ K\). It is proved that the result of \textit{M. S. Baouendi} and \textit{G. Métivier} [Am. J. Math. 104, 287-319 (1982; Zbl 0496.35026)]\ for the case \((k_ j=j)\) of an analytic vector (i.e. the analyticity of \(u\) in case \(P\) is of principal type and hypoelliptic) extends if and only if, for some \(A,\) \(k_{j+1}\leq Ak_ j\), \(j\in \mathbb{N}\). The same result applies also to a system of operators \(D_ 1,\dots,D_ n\).